Question

Consider damped harmonic oscillations. Let the coefficient of friction , be half of the one that yields critical damping.

(a) How many times larger is the period T than it would be for undamped motion (with γ = 0)?

          (b) Determine the ratio between the amplitudes of two successive swings to the same side. (I.e., find the maximum displacement of two successive oscillations to one side, and find the ratio of these two displacements)

Answer 1

a) for a damped harmonic oscillator the damped frequency

$\omega$_D = 1/2 ( 4$\omega$_o² - $\gamma$² )^1/2  

for critical damping $\gamma$ = 2$\omega$_o

the co-efficient of friction b/m is half of that for critical damping, hence

$\gamma$ = $\omega$_o

damped frequency  $\omega$_D = 1/2 ( 4$\omega$_o² - $\omega$_o² )^1/2 = 0.87$\omega$_o

Period T_D = 1.15 T_o  

b)

$x(t) = A e^{-t\gamma /2} Cos(\omega _Dt-\phi )$

The amplitude falls by $e^{-t\gamma /2}$ with time t

for two successive oscillations t = t and t +T_D

_{The ration of successive amplitudes =}  $e^{-T_D\gamma /2}$ , put T_D = 1.15T_o and $\gamma$ = $\omega$_o

= 0.027